Built with
Alectryon , running Lean4 v4.3.0. Bubbles (
) indicate interactive fragments: hover for details, tap to reveal contents. Use
Ctrl+↑ Ctrl+↓ to navigate,
Ctrl+🖱️ to focus. On Mac, use
⌘ instead of
Ctrl .
import Mathlib.Data.Real.EReal
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Bochner
import SBSProofs.Utils
import SBSProofs.PushForward
import SBSProofs.SteepestDirection
local macro_rules | ` ($ x ^ $ y) => ` (HPow.hPow $ x $ y)
open scoped RealInnerProductSpace
open BigOperators Finset ENNReal NNReal MeasureTheory
set_option trace.Meta.Tactic.simp.rewrite true
set_option maxHeartbeats 400000
/-
We defined measures μ and π (ν is considered as the standard Lebesgue measure) along with their densities (finite and non-zero on the entire space)
-/
variable {d : ℕ}
variable [MeasurableSpace (Vector ℝ d)] [MeasureSpace (Vector ℝ d)] [MeasureSpace ℝ]
variable (μ π ν : Measure (Vector ℝ d)) (dμ dπ : (Vector ℝ d) → ℝ≥ 0 ∞ )
/-
μ << π << ν, they both admit density w.r.t. ν.
-/
variable (_h1 : absolutely_continuous μ π) (_h2 : absolutely_continuous π ν)
example : absolutely_continuous μ ν := absolutely_continuous_trans _h1 _h2
variable (hμ : is_density μ ν dμ) (hπ : is_density π ν dπ) (mdμ : Measurable dμ) (mdπ : Measurable dπ) (hdμ : ∀ x, dμ x ≠ 0 ∧ dμ x ≠ ∞ ) (hdπ : ∀ x, dπ x ≠ 0 ∧ dπ x ≠ ∞ )
variable [IsProbabilityMeasure μ] [IsProbabilityMeasure π]
variable (h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet s)
/-
We define a RKHS of ((Vector ℝ d) → ℝ) functions.
-/
variable (H₀ : Set ((Vector ℝ d) → ℝ)) [NormedAddCommGroup ((Vector ℝ d) → ℝ)] [InnerProductSpace ℝ ((Vector ℝ d) → ℝ)] [s : RKHS H₀]
def positive_definite_kernel := ∀ (f : ℕ → Vector ℝ d → ℝ), (0 ≤ ∫ x in Set.univ, (∫ x' in Set.univ, (∑ i in range (d + 1 ), f i x * s.k x x' * f i x') ∂ μ) ∂ μ) ∧ (∫ x in Set.univ, (∫ x' in Set.univ, (∑ i in range (d + 1 ), f i x * s.k x x' * f i x') ∂ μ) ∂ μ = 0 ↔ ∀ x, ∀ i, f i x = 0 )
variable (h_kernel_positive : positive_definite_kernel μ H₀)
/- We define the product RKHS as a space of function on ℕ → (Vector ℝ d) to ℝ (vector-valued function in our Lean formalism). A function belongs to such a RKHS if f = (f_1, ..., f_d) and ∀ 1 ≤ i ≤ d, fᵢ ∈ H₀. -/
variable (H : Set (ℕ → (Vector ℝ d) → ℝ)) [NormedAddCommGroup (ℕ → (Vector ℝ d) → ℝ)] [InnerProductSpace ℝ (ℕ → (Vector ℝ d) → ℝ)]
/- ===============================KERNEL STEIN DISCREPANCY===============================-/
/-
Here, we prove that KSD(μ | π) is a valid discrepancy measure, that the Stein log Sobolev inequality holds, and the exponential convergence of SVGD.
-/
/- dk : x ↦ i ↦ y ↦ ∂xⁱ k(x, y) -/
variable (dk : (Vector ℝ d) → ℕ → (Vector ℝ d) → ℝ)
/- d_ln_π : i ↦ x ↦ ∂xⁱ ln (μ(x) / π(x)) -/
variable (d_ln_π : ℕ → (Vector ℝ d) → ℝ)
/-
Definition of the steepest direction ϕ
-/
variable (dϕ : ℕ → (Vector ℝ d) → ℝ)
/- We will use this assumption only when the function is trivially integrable (e.g. derivative of integrable functions). -/
variable (is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable f μ)
/-
d_ln_π_μ : i ↦ x ↦ ∂xⁱ ln (π(x) / μ(x))
-/
variable (d_ln_π_μ : ℕ → (Vector ℝ d) → ℝ)
/-
Simple derivative rule: if the derivative is 0 ∀x, then the function is constant.
-/
variable (hd_ln_π_μ : (∀ x, ∀ i, d_ln_π_μ i x = 0 ) → (∃ c, ∀ x, log (dμ x / dπ x) = c))
/-
dπ' : i ↦ x ↦ ∂xⁱ π(x)
-/
variable (dπ' : ℕ → (Vector ℝ d) → ℝ)
/-
Simple derivative rule: ∂xⁱ ln (π(x)) * π(x) = ∂xⁱ π(x).
-/
variable (hπ' : ∀ x, ∀ i, ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i x)
variable [Norm (Vector ℝ d)]
/--
Stein class of measure. f is in the Stein class of μ if, ∀i ∈ range (d + 1), lim_(‖x‖ → ∞) μ(x) * ϕ(x)ⁱ = 0.
-/
def SteinClass (f : ℕ → (Vector ℝ d) → ℝ) (dμ : (Vector ℝ d) → ℝ≥ 0 ∞ ) := ∀ x, tends_to_infty (fun (x : Vector ℝ d) ↦ ‖ x‖ ) → ∀ i, ENNReal.toReal (dμ x) * f i x = 0
/-
Kernel Stein Discrepancy
-/
variable (KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ)
/--
KSD(μ | π) = ⟪∇ln π/μ, Pμ ∇ln π/μ⟫_L²(μ). We assume here that KSD is also equal to ∫ x, ∑ l in range (d + 1), (d_ln_π l x * ϕ l x + dϕ l x) ∂μ.
-/
def is_ksd := KSD μ π = (∫ x in Set.univ, (∫ x' in Set.univ, (∑ i in range (d + 1 ), d_ln_π_μ i x * s.k x x' * d_ln_π_μ i x') ∂ μ) ∂ μ) ∧ (KSD μ π = ∫ x, ∑ l in range (d + 1 ), (d_ln_π l x * (ϕ_ μ H₀ dk d_ln_π) l x + dϕ l x) ∂ μ)
/-
KSD(μ | π) is originally defined as ‖ϕ^⋆‖²_H, it is therefore non-negative.
-/
variable (ksd_nn : 0 ≤ KSD μ π)
/-
ϕ is in the Stein class of π
-/
variable (hstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ)
/--
We show that, if ϕ is in the Stein class of π, KSD is a valid discrepancy measure i.e. μ = π ↔ KSD(μ | π) = 0.
-/
lemma KSD_is_valid_discrepancy (hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD) : μ = π ↔ KSD μ π = 0 :=
by
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ = π ↔ KSD μ π = 0
let ϕ := (ϕ_ μ H₀ dk d_ln_π)d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
μ = π ↔ KSD μ π = 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ = π ↔ KSD μ π = 0
constructor d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ = π ↔ KSD μ π = 0
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
-- μ = π ↦ KSD(μ | π) = 0.
intro h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
mp KSD μ π = 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
mp KSD μ π = 0
hksd.right d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
mp ∫ (x : Vector ℝ d), ∑ l in range (d + 1 ), (d_ln_π l x * ϕ_ μ H₀ dk d_ln_π l x + dϕ l x) ∂ μ = 0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
mp ∫ (x : Vector ℝ d), ∑ l in range (d + 1 ), (d_ln_π l x * ϕ_ μ H₀ dk d_ln_π l x + dϕ l x) ∂ μ = 0
-- ∑ i, f i + g i = ∑ i, f i + ∑ i, g i.
have split_sum : ∀ x, ∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) = (∑ l in range (d + 1 ), d_ln_π l x * ϕ l x) + (∑ l in range (d + 1 ), dϕ l x) := fun x ↦ sum_add_distribd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l x
mp ∫ (x : Vector ℝ d), ∑ l in range (d + 1 ), (d_ln_π l x * ϕ_ μ H₀ dk d_ln_π l x + dϕ l x) ∂ μ = 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l x
mp ∫ (x : Vector ℝ d), ∑ l in range (d + 1 ), (d_ln_π l x * ϕ_ μ H₀ dk d_ln_π l x + dϕ l x) ∂ μ = 0
split_sum [Meta.Tactic.simp.rewrite] split_sum:1000 , ∑ l in Finset.range (d + 1 ),
(d_ln_π l x * ϕ_ μ H₀ dk d_ln_π l x +
dϕ l x) ==> ∑ l in Finset.range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in Finset.range (d + 1 ), dϕ l x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l x
mp ∫ (x : Vector ℝ d), ∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x + ∑ l in range (d + 1 ), dϕ l x ∂ μ = 0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l x
mp ∫ (x : Vector ℝ d), ∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x + ∑ l in range (d + 1 ), dϕ l x ∂ μ = 0
-- Split the integral of sum into sum of integral.
have h1 : Integrable (fun x ↦ (∑ l in range (d + 1 ), d_ln_π l x * ϕ l x)) μ := is_integrable_H₀ _d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x
mp ∫ (x : Vector ℝ d), ∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x + ∑ l in range (d + 1 ), dϕ l x ∂ μ = 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
have h2 : Integrable (fun x ↦ (∑ l in range (d + 1 ), dϕ l x)) μ := is_integrable_H₀ _d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x
mp ∫ (x : Vector ℝ d), ∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x + ∑ l in range (d + 1 ), dϕ l x ∂ μ = 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x
mp ∫ (x : Vector ℝ d), ∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x + ∑ l in range (d + 1 ), dϕ l x ∂ μ = 0
integral_add (h1) h2 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x
mp ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ + ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ μ =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x
mp ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ + ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ μ =
0
-- Make the `Set.univ` appears for using the density later.
have int_univ : ∫ a, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ = ∫ a in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x
mp ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ + ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ μ =
0
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
simp [Meta.Tactic.simp.rewrite] @ MeasureTheory.Measure.restrict_univ:1000 , MeasureTheory.Measure.restrict μ Set.univ ==> μ
[Meta.Tactic.simp.rewrite] @ eq_self:1000 , ∫ (a : Vector ℝ d),
∑ x in Finset.range (d + 1 ), d_ln_π x a * ϕ_ μ H₀ dk d_ln_π x a ∂ μ =
∫ (a : Vector ℝ d), ∑ x in Finset.range (d + 1 ), d_ln_π x a * ϕ_ μ H₀ dk d_ln_π x a ∂ μ ==> True
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ + ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ μ =
0
int_univ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ μ =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ μ =
0
-- Replace μ by π in the integration.
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ μ =
0
h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ π +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ π +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
-- Replace by its density.
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ π +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
density_integration π ν dπ hπ (fun x ↦ (∑ l in range (d + 1 ), d_ln_π l x * ϕ l x)) Set.univ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
-- Get ENNReal.toReal (dπ x) in the sum (a * ∑ b = ∑ b * a).
have mul_dist : ∀ x, ENNReal.toReal (dπ x) * (∑ l in range (d + 1 ), (fun l ↦ d_ln_π l x * ϕ l x) l) = ∑ l in range (d + 1 ), (fun l ↦ d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x) := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
mp ∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
have mul_dist_sum : ∀ (a : ℝ), ∀ (f : ℕ → ℝ), (∑ i in range (d + 1 ), f i) * a = ∑ i in range (d + 1 ), f i * a := fun a f ↦ Finset.sum_muld : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist_sum : ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d + 1 ), f i) * a = ∑ i in range (d + 1 ), f i * a
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
intro x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist_sum : ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d + 1 ), f i) * a = ∑ i in range (d + 1 ), f i * ax : Vector ℝ d
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist_sum : ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d + 1 ), f i) * a = ∑ i in range (d + 1 ), f i * ax : Vector ℝ d
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mul_comm d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist_sum : ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d + 1 ), f i) * a = ∑ i in range (d + 1 ), f i * ax : Vector ℝ d
(∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l) * ENNReal.toReal (dπ x) =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist_sum : ∀ (a : ℝ) (f : ℕ → ℝ), (∑ i in range (d + 1 ), f i) * a = ∑ i in range (d + 1 ), f i * ax : Vector ℝ d
(∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l) * ENNReal.toReal (dπ x) =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ
∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
exact mul_dist_sum (ENNReal.toReal (dπ x)) (fun l ↦ d_ln_π l x * ϕ l x)
}
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mp ∫ (x : Vector ℝ d) in Set.univ,
ENNReal.toReal (dπ x) * ∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
mul_dist [Meta.Tactic.simp.rewrite] mul_dist:1000 , ENNReal.toReal (dπ x) *
∑ x_1 in Finset.range (d + 1 ),
d_ln_π x_1 x *
ϕ_ μ H₀ dk d_ln_π x_1
x ==> ∑ l in Finset.range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x) d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mp ∫ (x : Vector ℝ d) in Set.univ,
∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x * ENNReal.toReal (dπ x) ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)
mp ∫ (x : Vector ℝ d) in Set.univ,
∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x * ENNReal.toReal (dπ x) ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
-- Make the product ENNReal.toReal (dπ x) * d_ln_π i x appears to use the ln derivative rule.
have mul_comm : ∀ x, ∀ i, d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x := fun x i ↦ (mul_rotate (ENNReal.toReal (dπ x)) (d_ln_π i x) (ϕ i x)). symmd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ,
∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x * ENNReal.toReal (dπ x) ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ,
∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x * ENNReal.toReal (dπ x) ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
mul_comm [Meta.Tactic.simp.rewrite] mul_comm:1000 , d_ln_π x x✝ * ϕ_ μ H₀ dk d_ln_π x x✝ *
ENNReal.toReal (dπ x✝ ) ==> ENNReal.toReal (dπ x✝ ) * d_ln_π x x✝ * ϕ x x✝ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ,
∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x * ENNReal.toReal (dπ x) ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
, d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ,
∑ x_1 in range (d + 1 ), ENNReal.toReal (dπ x) * d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ,
∑ x_1 in range (d + 1 ), d_ln_π x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x * ENNReal.toReal (dπ x) ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
hπ' [Meta.Tactic.simp.rewrite] hπ':1000 , ENNReal.toReal (dπ x✝ ) * d_ln_π x x✝ ==> dπ' x x✝ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
-- Make the `Set.univ` appears to use the density.
have int_univ : ∫ a, ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ a in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i x
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
simp [Meta.Tactic.simp.rewrite] @ MeasureTheory.Measure.restrict_univ:1000 , MeasureTheory.Measure.restrict π Set.univ ==> π
[Meta.Tactic.simp.rewrite] @ eq_self:1000 , ∫ (a : Vector ℝ d), ∑ l in Finset.range (d + 1 ), dϕ l a ∂ π =
∫ (a : Vector ℝ d), ∑ l in Finset.range (d + 1 ), dϕ l a ∂ π ==> True
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
int_univ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π =
0
density_integration π ν dπ hπ (fun x ↦ (∑ l in range (d + 1 ), dϕ l x)) Set.univ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), dϕ l x ∂ ν =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), dϕ l x ∂ ν =
0
-- Use the integration by parts on the right-hand side integral.
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
∫ (x : Vector ℝ d) in Set.univ, ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), dϕ l x ∂ ν =
0
mv_integration_by_parts (fun x ↦ ENNReal.toReal (dπ x)) ϕ dπ' dϕ (hstein) d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
-∫ (x : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), dπ' i x * ϕ i x ∂ ν =
0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π split_sum : ∀ (x : Vector ℝ d),
∑ l in range (d + 1 ), (d_ln_π l x * ϕ l x + dϕ l x) =
∑ l in range (d + 1 ), d_ln_π l x * ϕ l x + ∑ l in range (d + 1 ), dϕ l xh1 : Integrable fun x => ∑ l in range (d + 1 ), d_ln_π l x * ϕ l x h2 : Integrable fun x => ∑ l in range (d + 1 ), dϕ l x int_univ✝ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μ =
∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), d_ln_π l a * ϕ l a ∂ μmul_dist : ∀ (x : Vector ℝ d),
ENNReal.toReal (dπ x) * ∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l =
∑ l in range (d + 1 ), (fun l => d_ln_π l x * ϕ l x) l * ENNReal.toReal (dπ x)mul_comm : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π i x * ϕ i x * ENNReal.toReal (dπ x) = ENNReal.toReal (dπ x) * d_ln_π i x * ϕ i xint_univ : ∫ (a : Vector ℝ d), ∑ l in range (d + 1 ), dϕ l a ∂ π = ∫ (a : Vector ℝ d) in Set.univ, ∑ l in range (d + 1 ), dϕ l a ∂ π
mp ∫ (x : Vector ℝ d) in Set.univ, ∑ x_1 in range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
-∫ (x : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), dπ' i x * ϕ i x ∂ ν =
0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mp μ = π → KSD μ π = 0
simp [Meta.Tactic.simp.rewrite] @ MeasureTheory.Measure.restrict_univ:1000 , MeasureTheory.Measure.restrict ν Set.univ ==> ν
[Meta.Tactic.simp.rewrite] @ add_right_neg:1000 , ∫ (x : Vector ℝ d),
∑ x_1 in Finset.range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν +
-∫ (x : Vector ℝ d), ∑ x_1 in Finset.range (d + 1 ), dπ' x_1 x * ϕ_ μ H₀ dk d_ln_π x_1 x ∂ ν ==> 0
[Meta.Tactic.simp.rewrite] @ eq_self:1000 , 0 = 0 ==> True
} d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ = π ↔ KSD μ π = 0
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
-- KSD(μ | π) = 0 ↦ μ = π.
intro h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : KSD μ π = 0
mpr μ = π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : KSD μ π = 0
mpr μ = π
hksd.left d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0
mpr μ = π
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0
mpr μ = π
at h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0
mpr μ = π
-- We use the fact that the kernel is positive-definite that implies that d_ln_π_μ = 0.
have d_ln_π_μ_eq_0 := (h_kernel_positive d_ln_π_μ). right.mp hd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0
mpr μ = π
-- Simple derivative rule: ∂x f x = 0 → f x = c
specialize hd_ln_π_μ d_ln_π_μ_eq_0 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 hd_ln_π_μ : ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
mpr μ = π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
rcases hd_ln_π_μ with ⟨c, h⟩ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
mpr.intro μ = π
-- We show that, since dμ x / dπ x ≠ 0 and finite, dμ x = ENNReal.ofReal (Real.exp c) * dπ x.
have dμ_propor : ∀ x, dμ x = ENNReal.ofReal (Real.exp c) * dπ x := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
mpr.intro μ = π
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
intro x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cx : Vector ℝ d
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
specialize h x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
have frac_neq_zero : dμ x / dπ x ≠ 0 := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c
dμ x / dπ x ≠ 0
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c
dμ x / dπ x ≠ 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c
dμ x / dπ x ≠ 0
have frac_pos : 0 < dμ x / dπ x := ENNReal.div_pos_iff.mpr ⟨(hdμ x). left, (hdπ x). right⟩d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_pos : 0 < dμ x / dπ x
dμ x / dπ x ≠ 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c
dμ x / dπ x ≠ 0
exact zero_lt_iff.mp frac_pos
}
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
have frac_finite : dμ x / dπ x ≠ ∞ := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0
dμ x / dπ x ≠ ⊤
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0
dμ x / dπ x ≠ ⊤
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0
dμ x / dπ x ≠ ⊤
by_contra h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝¹ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h✝ : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 h : dμ x / dπ x = ⊤
False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0
dμ x / dπ x ≠ ⊤
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝¹ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h✝ : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 h : dμ x / dπ x = ⊤
False
div_eq_top d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝¹ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h✝ : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 h : dμ x ≠ 0 ∧ dπ x = 0 ∨ dμ x = ⊤ ∧ dπ x ≠ ⊤
False
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝¹ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h✝ : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 h : dμ x ≠ 0 ∧ dπ x = 0 ∨ dμ x = ⊤ ∧ dπ x ≠ ⊤
False
at h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝¹ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h✝ : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 h : dμ x ≠ 0 ∧ dπ x = 0 ∨ dμ x = ⊤ ∧ dπ x ≠ ⊤
False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0
dμ x / dπ x ≠ ⊤
cases h with
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝¹ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h✝ : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 h : dμ x ≠ 0 ∧ dπ x = 0 ∨ dμ x = ⊤ ∧ dπ x ≠ ⊤
False
| inl hp => d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hp : dμ x ≠ 0 ∧ dπ x = 0
inl False
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hp : dμ x ≠ 0 ∧ dπ x = 0
inl False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hp : dμ x ≠ 0 ∧ dπ x = 0
inl False
rcases hp with ⟨ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hp : dμ x ≠ 0 ∧ dπ x = 0
inl False
hpl Warning: unused variable ` hpl` [linter.unusedVariables] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hp : dμ x ≠ 0 ∧ dπ x = 0
inl False
, hpr⟩ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hpl : dμ x ≠ 0 hpr : dπ x = 0
inl.intro False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hp : dμ x ≠ 0 ∧ dπ x = 0
inl False
exact (hdπ x). left hpr
}
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝¹ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h✝ : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 h : dμ x ≠ 0 ∧ dπ x = 0 ∨ dμ x = ⊤ ∧ dπ x ≠ ⊤
False
| inr hq => d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hq : dμ x = ⊤ ∧ dπ x ≠ ⊤
inr False
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hq : dμ x = ⊤ ∧ dπ x ≠ ⊤
inr False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hq : dμ x = ⊤ ∧ dπ x ≠ ⊤
inr False
rcases hq with ⟨hql, d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hq : dμ x = ⊤ ∧ dπ x ≠ ⊤
inr False
hqr Warning: unused variable ` hqr` [linter.unusedVariables] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hq : dμ x = ⊤ ∧ dπ x ≠ ⊤
inr False
⟩ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hql : dμ x = ⊤ hqr : dπ x ≠ ⊤
inr.intro False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 hq : dμ x = ⊤ ∧ dπ x ≠ ⊤
inr False
exact (hdμ x). right hql
}
}
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
have cancel_ln_exp : dμ x / dπ x = ENNReal.ofReal (Real.exp c) := cancel_ln_exp (dμ x / dπ x) frac_neq_zero frac_finite c hd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ x : Vector ℝ d h : log (dμ x / dπ x) = c frac_neq_zero : dμ x / dπ x ≠ 0 frac_finite : dμ x / dπ x ≠ ⊤ cancel_ln_exp : dμ x / dπ x = ENNReal.ofReal (Real.exp c)
dμ x = ENNReal.ofReal (Real.exp c) * dπ x
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c
∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
simp [← cancel_ln_exp, ENNReal.div_eq_inv_mul, mul_right_comm (dπ x)⁻¹ (dμ x) (dπ x), ENNReal.inv_mul_cancel (hdπ x). left (hdπ x). right] [Meta.Tactic.simp.rewrite] ← cancel_ln_exp:1000 , ENNReal.ofReal (Real.exp c) ==> dμ x / dπ x
[Meta.Tactic.simp.rewrite] @ ENNReal.div_eq_inv_mul:1000 , dμ x / dπ x ==> (dπ x)⁻¹ * dμ x
[Meta.Tactic.simp.rewrite] mul_right_comm (dπ x)⁻¹ (dμ x) (dπ x):1000 , (dπ x)⁻¹ * dμ x * dπ x ==> (dπ x)⁻¹ * dπ x * dμ x
[Meta.Tactic.simp.rewrite] ENNReal.inv_mul_cancel (hdπ x). left (hdπ x). right:1000 , (dπ x)⁻¹ * dπ x ==> 1
[Meta.Tactic.simp.rewrite] @ one_mul:1000 , 1 * dμ x ==> dμ x
[Meta.Tactic.simp.rewrite] @ eq_self:1000 , dμ x = dμ x ==> True
}
-- We show by cases that ENNReal.ofReal (Real.exp c) = 1. If it is ≠ 1, this implies a contradiction as dμ x = ENNReal.ofReal (Real.exp c) * dπ x and ∫⁻ x, dμ x ∂ν = 1.
have exp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
mpr.intro μ = π
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) = 1
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) = 1
by_cases hc : ENNReal.ofReal (Real.exp c) = 1 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) = 1
pos ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) = 1
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) = 1
pos ENNReal.ofReal (Real.exp c) = 1
assumption } d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ x
ENNReal.ofReal (Real.exp c) = 1
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
push_neg at hc d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
have univ_eq_one_μ : ∫⁻ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1
neg ENNReal.ofReal (Real.exp c) = 1
x Warning: unused variable ` x` [linter.unusedVariables] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1
neg ENNReal.ofReal (Real.exp c) = 1
in Set.univ, 1 ∂ μ = 1 := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1
neg ENNReal.ofReal (Real.exp c) = 1
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1
∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1
simp [Meta.Tactic.simp.rewrite] @ MeasureTheory.Measure.restrict_univ:1000 , MeasureTheory.Measure.restrict μ Set.univ ==> μ
[Meta.Tactic.simp.rewrite] @ MeasureTheory.lintegral_const:1000 , ∫⁻ (x : Vector ℝ d), 1 ∂ μ ==> 1 * ↑↑ μ Set.univ
[Meta.Tactic.simp.rewrite] @ MeasureTheory.measure_univ:1000 , ↑↑ μ Set.univ ==> 1
[Meta.Tactic.simp.rewrite] @ mul_one:1000 , 1 * 1 ==> 1
[Meta.Tactic.simp.rewrite] @ eq_self:1000 , 1 = 1 ==> True
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
have univ_eq_one_π : ∫⁻ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1
neg ENNReal.ofReal (Real.exp c) = 1
x Warning: unused variable ` x` [linter.unusedVariables] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1
neg ENNReal.ofReal (Real.exp c) = 1
in Set.univ, 1 ∂ π = 1 := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1
neg ENNReal.ofReal (Real.exp c) = 1
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1
∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
simp [Meta.Tactic.simp.rewrite] @ MeasureTheory.Measure.restrict_univ:1000 , MeasureTheory.Measure.restrict π Set.univ ==> π
[Meta.Tactic.simp.rewrite] @ MeasureTheory.lintegral_const:1000 , ∫⁻ (x : Vector ℝ d), 1 ∂ π ==> 1 * ↑↑ π Set.univ
[Meta.Tactic.simp.rewrite] @ MeasureTheory.measure_univ:1000 , ↑↑ π Set.univ ==> 1
[Meta.Tactic.simp.rewrite] @ mul_one:1000 , 1 * 1 ==> 1
[Meta.Tactic.simp.rewrite] @ eq_self:1000 , 1 = 1 ==> True
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
neg ENNReal.ofReal (Real.exp c) = 1
density_lintegration μ ν dμ hμ (fun d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
neg ENNReal.ofReal (Real.exp c) = 1
x Warning: unused variable ` x` [linter.unusedVariables] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ μ = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
neg ENNReal.ofReal (Real.exp c) = 1
↦ 1 ) Set.univ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
neg ENNReal.ofReal (Real.exp c) = 1
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
neg ENNReal.ofReal (Real.exp c) = 1
at univ_eq_one_μ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, dμ x * 1 ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1
neg ENNReal.ofReal (Real.exp c) = 1
dμ_propor [Meta.Tactic.simp.rewrite] dμ_propor:1000 , dμ x ==> ENNReal.ofReal (Real.exp c) * dπ x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x * 1 ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
] at univ_eq_one_μ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x * 1 ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x * 1 ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
mul_one [Meta.Tactic.simp.rewrite] @ mul_one:1000 , ENNReal.ofReal (Real.exp c) * dπ x * 1 ==> ENNReal.ofReal (Real.exp c) * dπ x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
] at univ_eq_one_μ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
density_lintegration π ν dπ hπ (fun d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
x Warning: unused variable ` x` [linter.unusedVariables] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, 1 ∂ π = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
↦ 1 ) Set.univ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ∂ ν = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ∂ ν = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
at univ_eq_one_π d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ∂ ν = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x * 1 ∂ ν = 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
mul_one [Meta.Tactic.simp.rewrite] @ mul_one:1000 , dπ x * 1 ==> dπ x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
] at univ_eq_one_π d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
lintegral_const_mul (ENNReal.ofReal (Real.exp c)) (mdπ), d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ENNReal.ofReal (Real.exp c) * ∫⁻ (a : Vector ℝ d) in Set.univ, dπ a ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
univ_eq_one_π, d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ENNReal.ofReal (Real.exp c) * 1 = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ∫⁻ (x : Vector ℝ d) in Set.univ, ENNReal.ofReal (Real.exp c) * dπ x ∂ ν = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
mul_one d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ENNReal.ofReal (Real.exp c) = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ENNReal.ofReal (Real.exp c) = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
at univ_eq_one_μ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ENNReal.ofReal (Real.exp c) = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg ENNReal.ofReal (Real.exp c) = 1
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
exfalso d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ENNReal.ofReal (Real.exp c) ≠ 1 univ_eq_one_μ : ENNReal.ofReal (Real.exp c) = 1 univ_eq_one_π : ∫⁻ (x : Vector ℝ d) in Set.univ, dπ x ∂ ν = 1
neg.h False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xhc : ¬ ENNReal.ofReal (Real.exp c) = 1
neg ENNReal.ofReal (Real.exp c) = 1
exact hc univ_eq_one_μ
}
}
-- We rewrite μ = π as ∀s, ∫⁻ x in s, dμ ∂ν = ∀s, ∫⁻ x in s, dπ ∂ν and use dμ = 1 * dπ.
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1
mpr.intro μ = π
exp_c_eq_one [Meta.Tactic.simp.rewrite] exp_c_eq_one:1000 , ENNReal.ofReal (Real.exp c) ==> 1 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1
mpr.intro μ = π
, d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = 1 * dπ x
mpr.intro μ = π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cdμ_propor : ∀ (x : Vector ℝ d), dμ x = ENNReal.ofReal (Real.exp c) * dπ xexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1
mpr.intro μ = π
one_mul [Meta.Tactic.simp.rewrite] @ one_mul:1000 , 1 * dπ x ==> dπ x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ x
mpr.intro μ = π
] at dμ_propor d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ x
mpr.intro μ = π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
ext s _hs d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ↑↑ μ s = ↑↑ π s
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ↑↑ μ s = ↑↑ π s
← set_lintegral_one s,d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, 1 ∂ μ = ↑↑ π s
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ↑↑ μ s = ↑↑ π s
← set_lintegral_one sd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, 1 ∂ μ = ∫⁻ (x : Vector ℝ d) in s, 1 ∂ π
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, 1 ∂ μ = ∫⁻ (x : Vector ℝ d) in s, 1 ∂ π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, 1 ∂ μ = ∫⁻ (x : Vector ℝ d) in s, 1 ∂ π
density_lintegration μ ν dμ hμ, d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ∂ ν = ∫⁻ (x : Vector ℝ d) in s, 1 ∂ π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, 1 ∂ μ = ∫⁻ (x : Vector ℝ d) in s, 1 ∂ π
density_lintegration π ν dπ hπ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ∂ ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ∂ ν
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ∂ ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ∂ ν
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
mpr KSD μ π = 0 → μ = π
simp_rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ∂ ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ∂ ν
mul_one [Meta.Tactic.simp.rewrite] @ mul_one:1000 , dμ x * 1 ==> dμ x
[Meta.Tactic.simp.rewrite] @ mul_one:1000 , dπ x * 1 ==> dπ x d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ∂ ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ∂ ν
, d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, dμ x ∂ ν = ∫⁻ (x : Vector ℝ d) in s, dπ x ∂ ν
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s✝ : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ, dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h✝ : ∫ (x : Vector ℝ d) in Set.univ,
∫ (x' : Vector ℝ d) in Set.univ, ∑ i in range (d + 1 ), d_ln_π_μ i x * RKHS.k H₀ x x' * d_ln_π_μ i x' ∂ μ ∂ μ =
0 d_ln_π_μ_eq_0 : ∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 c : ℝ h : ∀ (x : Vector ℝ d), log (dμ x / dπ x) = cexp_c_eq_one : ENNReal.ofReal (Real.exp c) = 1 dμ_propor : ∀ (x : Vector ℝ d), dμ x = dπ xs : Set (Vector ℝ d) _hs : MeasurableSet s
mpr.intro.h ∫⁻ (x : Vector ℝ d) in s, dμ x * 1 ∂ ν = ∫⁻ (x : Vector ℝ d) in s, dπ x * 1 ∂ ν
dμ_propor [Meta.Tactic.simp.rewrite] dμ_propor:1000 , dμ x ==> dπ x
[Meta.Tactic.simp.rewrite] @ eq_self:1000 , ∫⁻ (x : Vector ℝ d) in s, dπ x ∂ ν = ∫⁻ (x : Vector ℝ d) in s, dπ x ∂ ν ==> True ]
}
/-
Basic proprieties of KL.
-/
variable (hkl_iff : μ = π ↔ KL μ dμ dπ = 0 ) (hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ)
/--
We show that μ ≠ π → 0 < KSD μ π (trivial using *KSD_is_valid_discrepancy*).
-/
lemma μ_neq_π_imp_ksd_nn (hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD) : μ ≠ π → 0 < KSD μ π :=
by
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ ≠ π → 0 < KSD μ π
intro h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π
0 < KSD μ π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ ≠ π → 0 < KSD μ π
by_contra h2 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : ¬ 0 < KSD μ π
False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ ≠ π → 0 < KSD μ π
push_neg at h2 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0
False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ ≠ π → 0 < KSD μ π
have split_le := LE.le.lt_or_eq h2d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 split_le : KSD μ π < 0 ∨ KSD μ π = 0
False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
μ ≠ π → 0 < KSD μ π
cases split_le with
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 split_le : KSD μ π < 0 ∨ KSD μ π = 0
False
| inl lt => d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 lt : KSD μ π < 0
inl False
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 lt : KSD μ π < 0
inl False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 lt : KSD μ π < 0
inl False
linarith }
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 split_le : KSD μ π < 0 ∨ KSD μ π = 0
False
| inr eq => d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 eq : KSD μ π = 0
inr False
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 eq : KSD μ π = 0
inr False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 eq : KSD μ π = 0
inr False
have μ_eq_π := (KSD_is_valid_discrepancy μ π ν dμ dπ hμ hπ mdπ hdμ hdπ H₀ h_kernel_positive dk d_ln_π dϕ is_integrable_H₀ d_ln_π_μ hd_ln_π_μ dπ' hπ' KSD hstein hksd). mpr eqd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 eq : KSD μ π = 0 μ_eq_π : μ = π
inr False
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD h : μ ≠ π h2 : KSD μ π ≤ 0 eq : KSD μ π = 0
inr False
exact h μ_eq_π
}
/--
We show that it exists a finite and positive θ such that KL(μ || π) ≤ (1 / (2θ)) * KSD(μ | π)
-/
theorem Stein_log_Sobolev (hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD) : ∃ θ > 0 , ((θ ≠ ∞ ) ∧ (KL μ dμ dπ ≤ (1 / (2 * θ)) * ENNReal.ofReal (KSD μ π))) :=
by
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
let d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
ϕ Warning: unused variable ` ϕ` [linter.unusedVariables] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
:= (ϕ_ μ H₀ dk d_ln_π) d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ
∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
by_cases h : μ = π d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
-- μ = π → KSD μ π = 0
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
(KSD_is_valid_discrepancy μ π ν dμ dπ hμ hπ mdπ hdμ hdπ H₀ h_kernel_positive dk d_ln_π dϕ is_integrable_H₀ d_ln_π_μ hd_ln_π_μ dπ' hπ' KSD hstein hksd). mp h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal 0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal 0
-- μ = π → KL μ π = 0
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal 0
hkl_iff.mp h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ 0 ≤ 1 / (2 * θ) * ENNReal.ofReal 0
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ 0 ≤ 1 / (2 * θ) * ENNReal.ofReal 0
-- Use any θ > 0 ∧ θ ≠ ∞
use 1 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
h 1 > 0 ∧ 1 ≠ ⊤ ∧ 0 ≤ 1 / (2 * 1 ) * ENNReal.ofReal 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ = π
pos ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
simp [Meta.Tactic.simp.rewrite] @ gt_iff_lt:1000 , 1 > 0 ==> 0 < 1
[Meta.Tactic.simp.rewrite] @ zero_lt_one:1000 , 0 < 1 ==> True
[Meta.Tactic.simp.rewrite] @ mul_one:1000 , 2 * 1 ==> 2
[Meta.Tactic.simp.rewrite] @ one_div:1000 , 1 / 2 ==> 2 ⁻¹
[Meta.Tactic.simp.rewrite] ENNReal.ofReal_zero:1000 , ENNReal.ofReal 0 ==> 0
[Meta.Tactic.simp.rewrite] @ mul_zero:1000 , 2 ⁻¹ * 0 ==> 0
[Meta.Tactic.simp.rewrite] @ le_refl:1000 , 0 ≤ 0 ==> True
[Meta.Tactic.simp.rewrite] and_self:1000 , True ∧ True ==> True
[Meta.Tactic.simp.rewrite] and_self:1000 , True ∧ True ==> True
} d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ¬ μ = π
neg ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD
∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ¬ μ = π
neg ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
-- μ ≠ π
push_neg at h d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
neg ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
-- Let θ = KSD(μ | π) / (2 KL(μ || π)
use ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0 ∧
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤ ∧
KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ¬ μ = π
neg ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
constructor d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ¬ μ = π
neg ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
-- We show that 0 < KSD(μ | π) / (2 KL(μ || π) by showing that 0 < KSD(μ | π) and 2 KL(μ || π) ≠ ∞ (as both are non-negative).
have imp_lt : (0 < KSD μ π) ∧ ((2 * KL μ dμ dπ) ≠ ∞ ) → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
simp [Meta.Tactic.simp.rewrite] @ ne_eq:1000 , 2 * KL μ dμ dπ ≠ ⊤ ==> ¬ 2 * KL μ dμ dπ = ⊤
[Meta.Tactic.simp.rewrite] @ ENNReal.div_pos_iff:1000 , 0 <
ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ==> ENNReal.ofReal (KSD μ π) ≠ 0 ∧ 2 * KL μ dμ dπ ≠ ⊤
[Meta.Tactic.simp.rewrite] @ ne_eq:1000 , ENNReal.ofReal (KSD μ π) ≠ 0 ==> ¬ ENNReal.ofReal (KSD μ π) = 0
[Meta.Tactic.simp.rewrite] @ ENNReal.ofReal_eq_zero:1000 , ENNReal.ofReal (KSD μ π) = 0 ==> KSD μ π ≤ 0
[Meta.Tactic.simp.rewrite] @ not_le:1000 , ¬ KSD μ π ≤ 0 ==> 0 < KSD μ π
[Meta.Tactic.simp.rewrite] @ imp_self:1000 , 0 < KSD μ π ∧ ¬ 2 * KL μ dμ dπ = ⊤ →
0 < KSD μ π ∧ ¬ 2 * KL μ dμ dπ = ⊤ ==> True
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
apply imp_lt d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π imp_lt : 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
h.left 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
constructor d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π imp_lt : 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
h.left.left 0 < KSD μ π
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π imp_lt : 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
h.left.left 0 < KSD μ π
-- We use *μ_neq_π_imp_ksd_nn* as μ ≠ π.
exact μ_neq_π_imp_ksd_nn μ π ν dμ dπ hμ hπ mdπ hdμ hdπ H₀ h_kernel_positive dk d_ln_π dϕ is_integrable_H₀ d_ln_π_μ hd_ln_π_μ dπ' hπ' KSD ksd_nn hstein hksd h
} d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π imp_lt : 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
h.left.right 2 * KL μ dμ dπ ≠ ⊤
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) > 0
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π imp_lt : 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
h.left.right 2 * KL μ dμ dπ ≠ ⊤
-- KL is finite (in our framework, as μ << π << ν).
exact mul_ne_top ( d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π imp_lt : 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
h.left.right 2 * KL μ dμ dπ ≠ ⊤
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π imp_lt : 0 < KSD μ π ∧ 2 * KL μ dμ dπ ≠ ⊤ → 0 < ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ)
2 ≠ ⊤
simp ) (ofReal_ne_top)
}
} d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.right ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤ ∧
KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : ¬ μ = π
neg ∃ θ > 0 , θ ≠ ⊤ ∧ KL μ dμ dπ ≤ 1 / (2 * θ) * ENNReal.ofReal (KSD μ π)
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.right ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤ ∧
KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
-- μ ≠ π → KL(μ || π) ≠ 0
have KL_neq_0 : KL μ dμ dπ ≠ 0 := Iff.mp zero_lt_iff (hkl_diff.mp h)d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤ ∧
KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.right ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤ ∧
KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
constructor d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.right ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤ ∧
KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤
-- We show that KSD(μ | π) / (2 KL(μ || π) ≠ ∞ by showing that KSD(μ | π) ≠ ∞ and 2 (KL(μ || π))⁻¹ ≠ ∞ (as both are non-negative).
have div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ := rfld : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
h.right.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤
rw [ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
h.right.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤
div_as_inv d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
h.right.left ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ ≠ ⊤
] d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹
h.right.left ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ ≠ ⊤
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤
have enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ∞ := ofReal_ne_topd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
h.right.left ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ ≠ ⊤
-- We show that (KL(μ || π))⁻¹ ≠ ∞ as KL(μ || π) ≠ 0
have inv_KL_finite : (2 * KL μ dμ dπ)⁻¹ ≠ ∞ := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
h.right.left ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ ≠ ⊤
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
(2 * KL μ dμ dπ)⁻¹ ≠ ⊤
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
(2 * KL μ dμ dπ)⁻¹ ≠ ⊤
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
(2 * KL μ dμ dπ)⁻¹ ≠ ⊤
have neq_zero : 2 * KL μ dμ dπ ≠ 0 := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
(2 * KL μ dμ dπ)⁻¹ ≠ ⊤
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
2 * KL μ dμ dπ ≠ 0
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
2 * KL μ dμ dπ ≠ 0
simp [Meta.Tactic.simp.rewrite] @ ne_eq:1000 , 2 * KL μ dμ dπ ≠ 0 ==> ¬ 2 * KL μ dμ dπ = 0
[Meta.Tactic.simp.rewrite] @ mul_eq_zero:1000 , 2 * KL μ dμ dπ = 0 ==> 2 = 0 ∨ KL μ dμ dπ = 0
[Meta.Tactic.simp.rewrite] @ OfNat.ofNat_ne_zero:1000 , 2 = 0 ==> False
[Meta.Tactic.simp.rewrite] false_or:1000 , False ∨ KL μ dμ dπ = 0 ==> KL μ dμ dπ = 0 d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
¬ KL μ dμ dπ = 0
; d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
¬ KL μ dμ dπ = 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
2 * KL μ dμ dπ ≠ 0
exact KL_neq_0 }
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 div_as_inv : ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) = ENNReal.ofReal (KSD μ π) * (2 * KL μ dμ dπ)⁻¹ enn_KSD_finite : ENNReal.ofReal (KSD μ π) ≠ ⊤
(2 * KL μ dμ dπ)⁻¹ ≠ ⊤
exact inv_ne_top.mpr neq_zero
}
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.left ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤
exact mul_ne_top enn_KSD_finite inv_KL_finite
} d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.right KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π
h.right ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ) ≠ ⊤ ∧
KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.right KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
-- As μ ≠ π, 0 < KSD(μ | π) and thus, KSD(μ | π) ≠ 0.
have enn_KSD_neq_0 : ENNReal.ofReal (KSD μ π) ≠ 0 := d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.right KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
by d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
ENNReal.ofReal (KSD μ π) ≠ 0
{ d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
ENNReal.ofReal (KSD μ π) ≠ 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
ENNReal.ofReal (KSD μ π) ≠ 0
have KSD_ge_0 := μ_neq_π_imp_ksd_nn μ π ν dμ dπ hμ hπ mdπ hdμ hdπ H₀ h_kernel_positive dk d_ln_π dϕ is_integrable_H₀ d_ln_π_μ hd_ln_π_μ dπ' hπ' KSD ksd_nn hstein hksd hd : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 KSD_ge_0 : 0 < KSD μ π
ENNReal.ofReal (KSD μ π) ≠ 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
ENNReal.ofReal (KSD μ π) ≠ 0
have enn_KSD_ge_0 := Iff.mpr ofReal_pos KSD_ge_0d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 KSD_ge_0 : 0 < KSD μ πenn_KSD_ge_0 : 0 < ENNReal.ofReal (KSD μ π)
ENNReal.ofReal (KSD μ π) ≠ 0
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
ENNReal.ofReal (KSD μ π) ≠ 0
exact Iff.mp zero_lt_iff enn_KSD_ge_0
}
-- We show that KL(μ || π) is in fact equal to 1 / (2 * KSD (μ | π) / (2 * KL(μ || π)) * KSD (μ | π)
apply Eq.le d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0 enn_KSD_neq_0 : ENNReal.ofReal (KSD μ π) ≠ 0
h.right.right.a KL μ dμ dπ = 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
d : ℕ inst✝⁹ : MeasurableSpace (Vector ℝ d) inst✝⁸ : MeasureSpace (Vector ℝ d) inst✝⁷ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝⁶ : IsProbabilityMeasure μ inst✝⁵ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁴ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝³ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝² : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝¹ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ hksd : is_ksd μ π H₀ dk d_ln_π dϕ d_ln_π_μ KSD ϕ := ϕ_ μ H₀ dk d_ln_π : ℕ → Vector ℝ d → ℝ h : μ ≠ π KL_neq_0 : KL μ dμ dπ ≠ 0
h.right.right KL μ dμ dπ ≤ 1 / (2 * (ENNReal.ofReal (KSD μ π) / (2 * KL μ dμ dπ))) * ENNReal.ofReal (KSD μ π)
exact ennreal_quo_eq (ofReal_ne_top) (enn_KSD_neq_0)
}
}
}
/-
In this sub-section, we define the flow of measures μ_t:
μ_t : ℝ≥0 → Measure (Vector ℝ d)
t ↦ T_t#μ, where T_t is the trajectories associated with ϕ(μ_t t), the steepest direction to update μ_t t for minimizing ∂t KL(μ_t t || π).
We also define everything that we need to use previous results with each measures given by μ_t.
-/
variable (μ_t : ℝ≥ 0 → Measure (Vector ℝ d)) (dμ_t : ℝ≥ 0 → (Vector ℝ d → ℝ≥ 0 ∞ )) (hμ_t : ∀ t, is_density (μ_t t) ν (dμ_t t)) (h_prob : ∀ t, IsProbabilityMeasure (μ_t t))
variable (hdμ_t :∀ t, ∀ (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ )
/-
d_KL_t : t ↦ ∂t KL(μ_t t || π)
-/
variable (d_KL_t : ℝ≥ 0 → ℝ)
variable (ksd_nn_t : ∀ t, 0 ≤ KSD (μ_t t) π)
/-
Our definition of the Gronwall's lemma.
-/
variable [MeasureSpace ℝ≥ 0 ] [NormedAddCommGroup ℝ≥ 0 ∞ ] [NormedSpace ℝ ℝ≥ 0 ∞ ] [LocallyFiniteOrder ℝ≥ 0 ]
variable (gronwall : ∀ (ψ : ℝ≥ 0 → ℝ), ∀ t > 0 , d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) → KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ s in Icc 0 t, ψ s))
/-
We assume here that ∂t KL(μ_t t || π) = - KSD(μ_t t | π) (showed in the paper).
-/
variable (dkl_ksd : ∀ t, d_KL_t t ≤ - KSD (μ_t t) π)
/-
As showed in the paper (and reminded above), t ↦ KL(μ_t t || π) is decreasing.
-/
variable (kl_decreasing : ∀ t, ∀ t', t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπ) (kl_finite : ∀ t, KL (μ_t t) (dμ_t t) dπ ≠ ∞ )
/--
If t ↦ KL(μ_t t || π) is bounded from below by a strictly positive constant, it means that μ_t cannot be made arbitrary close to π and thus that t ↦ KSD(μ_t | π) can be bounded by a striclty positive constant. Admitted here; we plan on formally show it in the future.
-/
lemma KL_bounded_imp_bounded_KSD Warning: declaration uses ' sorry ' (α : ℝ≥ 0 ∞ ) (hα : 0 < α) (hkl : ∀ t, α < KL (μ_t t) (dμ_t t) dπ) : ∃ (β : ℝ≥ 0 ∞ ), (∀ t, β < ENNReal.ofReal (KSD (μ_t t) π)) ∧ (0 < β) ∧ (β ≠ ∞ ) := by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ α : ℝ≥ 0 ∞ hα : 0 < αhkl : ∀ (t : ℝ≥ 0 ), α < KL (μ_t t) (dμ_t t) dπ
∃ β, (∀ (t : ℝ≥ 0 ), β < ENNReal.ofReal (KSD (μ_t t) π)) ∧ 0 < β ∧ β ≠ ⊤
sorry
/--
Squeeze theorem: ∀t, 0 ≤ KL(μ_t t || π) ≤ f(t) ∧ lim_(t → ∞) f(t) = 0 → lim_(t → ∞) KL(μ_t t || π) = 0.
-/
lemma squeeze_th_KL Warning: declaration uses ' sorry ' (f : ℝ≥ 0 → ℝ≥ 0 ∞ ) (h : limit f 0 ) : (∀ t> 0 , KL (μ_t t) (dμ_t t) dπ ≤ f t) → limit (fun t ↦ KL (μ_t t) (dμ_t t) dπ) 0 := by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ f : ℝ≥ 0 → ℝ≥ 0 ∞ h : limit f 0
(∀ t > 0 , KL (μ_t t) (dμ_t t) dπ ≤ f t) → limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
sorry
/--
We show the convergence of SVGD i.e. lim_(t → ∞) KL (μ_t || π) = 0.
-/
theorem convergence_SVGD : limit (fun t ↦ KL (μ_t t) (dμ_t t) dπ) 0 :=
by
-- As t ↦ KL (μ_t || π) is decreasing and bound from below, it admits a limit l ≥ 0.
have admits_limit := decreasing_bounded_function_limit (fun t ↦ KL (μ_t t) (dμ_t t) dπ) 0 (d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤
∀ (x : ℝ≥ 0 ), 0 ≤ (fun t => KL (μ_t t) (dμ_t t) dπ) x
simp [Meta.Tactic.simp.rewrite] @ zero_le:1000 , 0 ≤ KL (μ_t x) (dμ_t x) dπ ==> True
[Meta.Tactic.simp.rewrite] @ forall_const:1000 , ℝ≥ 0 → True ==> True ) kl_decreasing d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ admits_limit : ∃ α, limit (fun t => KL (μ_t t) (dμ_t t) dπ) α ∧ 0 ≤ α ∧ ∀ (x : ℝ≥ 0 ), α < (fun t => KL (μ_t t) (dμ_t t) dπ) x
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
rcases admits_limit with ⟨l, lim, _lim_pos, KL_bounded⟩ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) x
intro.intro.intro limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
-- We proceed by cases on the value of l. If l = 0, the proof is finished. Otherwise, we proceed by contradiction by showing that l ≠ 0 → l = 0.
by_cases hl : 0 = l d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 = l
pos limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 = l
pos limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 = l
pos limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
rwa [ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 = l
pos limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
hl d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 = l
pos limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 = l
pos limit (fun t => KL (μ_t t) (dμ_t t) dπ) l
} d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤
limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
exfalso d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg.h False
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
push_neg at hl d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ l
neg.h False
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
have lim_nn : 0 < l := Iff.mpr zero_lt_iff (Ne.symm hl)d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < l
neg.h False
-- We use the *KL_bounded_imp_bounded_KSD* to extract 0 < γ < KSD(μ_t | π)
have KSD_bounded := KL_bounded_imp_bounded_KSD π dπ KSD μ_t dμ_t l lim_nn KL_boundedd : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lKSD_bounded : ∃ β, (∀ (t : ℝ≥ 0 ), β < ENNReal.ofReal (KSD (μ_t t) π)) ∧ 0 < β ∧ β ≠ ⊤
neg.h False
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
rcases KSD_bounded with ⟨γ, KSD_bounded, γ_nn, γ_finite⟩ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
neg.h.intro.intro.intro False
-- We use the fact that KSD is bounded to specialize the log Sobolev inequality with λ = KSD(μ_t | π) / (2 * KL(μ_t || π)).
have log_Sobolev : ∀ t, KL (μ_t t) (dμ_t t) dπ = 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
neg.h.intro.intro.intro False
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
intro t d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
have enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ∞ := ofReal_ne_topd : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
have enn_KSD_nn : ENNReal.ofReal (KSD (μ_t t) π) ≠ 0 := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
have t : 0 < ENNReal.ofReal (KSD (μ_t t) π) := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
0 < ENNReal.ofReal (KSD (μ_t t) π)
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
0 < ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
0 < ENNReal.ofReal (KSD (μ_t t) π)
specialize KSD_bounded t d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤ KSD_bounded : γ < ENNReal.ofReal (KSD (μ_t t) π)
0 < ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
0 < ENNReal.ofReal (KSD (μ_t t) π)
exact gt_trans KSD_bounded γ_nn
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ t : ℝ≥ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
exact zero_lt_iff.mp t
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
exact ennreal_quo_eq enn_KSD_finite enn_KSD_nn
}
-- We prove that γ/2KL(μ_0 || π) < KSD(μ_t | π) / 2KL(μ_t | π) (we plan on using it with the Stein log-Sobolev inequality).
have gamma_star := decrease_bound (fun t ↦ KL (μ_t t) (dμ_t t) dπ) (fun t ↦ ENNReal.ofReal (KSD (μ_t t) π)) kl_decreasing (fun x ↦ Iff.mp zero_lt_iff (gt_trans (KL_bounded x) lim_nn)) kl_finite γ KSD_boundedd : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
neg.h.intro.intro.intro False
-- γ/2KL(μ_0 || π) ≠ 0
have gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
neg.h.intro.intro.intro False
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
have two_KL_finite : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ∞ := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤
rw [ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤
two_mul d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
KL (μ_t 0 ) (dμ_t 0 ) dπ + KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
KL (μ_t 0 ) (dμ_t 0 ) dπ + KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤
exact add_ne_top.mpr ⟨kl_finite 0 , kl_finite 0 ⟩
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
have inv_two_KL_neq : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ 0 := ENNReal.inv_ne_zero.mpr two_KL_finited : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)two_KL_finite : 2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤ inv_two_KL_neq : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
have γ_neq : γ ≠ 0 := Iff.mp zero_lt_iff γ_nnd : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)two_KL_finite : 2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤ inv_two_KL_neq : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ 0 γ_neq : γ ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
rw [ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)two_KL_finite : 2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤ inv_two_KL_neq : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ 0 γ_neq : γ ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
ENNReal.div_eq_inv_mul d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)two_KL_finite : 2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤ inv_two_KL_neq : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ 0 γ_neq : γ ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ * γ ≠ 0
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)two_KL_finite : 2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ ⊤ inv_two_KL_neq : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ 0 γ_neq : γ ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ * γ ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
exact mul_ne_zero inv_two_KL_neq γ_neq
}
-- γ/2KL(μ_0 || π) ≠ ∞
have gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ∞ := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
neg.h.intro.intro.intro False
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
rw [ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
ENNReal.div_eq_inv_mul d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ * γ ≠ ⊤
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ * γ ≠ ⊤
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
have KL_finite : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ ∞ := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ * γ ≠ ⊤
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ ⊤
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ ⊤
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ ⊤
have tmp : (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ ⊤
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
rw [ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
two_mul d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
KL (μ_t 0 ) (dμ_t 0 ) dπ + KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
KL (μ_t 0 ) (dμ_t 0 ) dπ + KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
simp [Meta.Tactic.simp.rewrite] @ ne_eq:1000 , KL (μ_t 0 ) (dμ_t 0 ) dπ + KL (μ_t 0 ) (dμ_t 0 ) dπ ≠
0 ==> ¬ KL (μ_t 0 ) (dμ_t 0 ) dπ + KL (μ_t 0 ) (dμ_t 0 ) dπ = 0
[Meta.Tactic.simp.rewrite] @ add_eq_zero:1000 , KL (μ_t 0 ) (dμ_t 0 ) dπ + KL (μ_t 0 ) (dμ_t 0 ) dπ =
0 ==> KL (μ_t 0 ) (dμ_t 0 ) dπ = 0 ∧ KL (μ_t 0 ) (dμ_t 0 ) dπ = 0
[Meta.Tactic.simp.rewrite] and_self:1000 , KL (μ_t 0 ) (dμ_t 0 ) dπ = 0 ∧
KL (μ_t 0 ) (dμ_t 0 ) dπ = 0 ==> KL (μ_t 0 ) (dμ_t 0 ) dπ = 0 d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
¬ KL (μ_t 0 ) (dμ_t 0 ) dπ = 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
2 * KL (μ_t 0 ) (dμ_t 0 ) dπ ≠ 0
exact Iff.mp zero_lt_iff (gt_trans (KL_bounded 0 ) lim_nn)
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
(2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)⁻¹ ≠ ⊤
exact inv_ne_top.mpr tmp
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0
γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
exact mul_ne_top KL_finite γ_finite
}
-- Using the Stein log-Sobolev and the gamma_star inequality, we are able to show that, ∀t, KL(μ_t || π) < 1/(2*γ/2KL(μ_0 || π)) * KSD(μ_t | π).
have bounded_log_Sobolev : ∀ t, KL (μ_t t) (dμ_t t) dπ < (1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) )) * ENNReal.ofReal (KSD (μ_t t) π) := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
neg.h.intro.intro.intro False
by
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
intro t d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
specialize log_Sobolev t d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
specialize gamma_star t d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
have le_quotient : ∀ (a b : ℝ≥ 0 ∞ ), a ≠ 0 → a < b → 1 / (2 * b) < 1 / (2 * a) :=
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
by
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
∀ (a b : ℝ≥ 0 ∞ ), a ≠ 0 → a < b → 1 / (2 * b) < 1 / (2 * a)
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
∀ (a b : ℝ≥ 0 ∞ ), a ≠ 0 → a < b → 1 / (2 * b) < 1 / (2 * a)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
∀ (a b : ℝ≥ 0 ∞ ), a ≠ 0 → a < b → 1 / (2 * b) < 1 / (2 * a)
intro a b _ha h d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
1 / (2 * b) < 1 / (2 * a)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
∀ (a b : ℝ≥ 0 ∞ ), a ≠ 0 → a < b → 1 / (2 * b) < 1 / (2 * a)
rw[ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
1 / (2 * b) < 1 / (2 * a)
one_div (2 * a), d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
1 / (2 * b) < (2 * a)⁻¹
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
1 / (2 * b) < 1 / (2 * a)
one_div (2 * b) d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
(2 * b)⁻¹ < (2 * a)⁻¹
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
(2 * b)⁻¹ < (2 * a)⁻¹
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
∀ (a b : ℝ≥ 0 ∞ ), a ≠ 0 → a < b → 1 / (2 * b) < 1 / (2 * a)
have h : 2 * a < 2 * b := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
(2 * b)⁻¹ < (2 * a)⁻¹
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
2 * a < 2 * b
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
2 * a < 2 * b
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
2 * a < 2 * b
repeat d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
2 * a < 2 * b
rw[ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
2 * a < 2 * b
two_mul d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
a + a < b + b
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
a + a < b + b
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) a, b : ℝ≥ 0 ∞ _ha : a ≠ 0 h : a < b
2 * a < 2 * b
exact ENNReal.add_lt_add h h
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)
∀ (a b : ℝ≥ 0 ∞ ), a ≠ 0 → a < b → 1 / (2 * b) < 1 / (2 * a)
exact Iff.mpr ENNReal.inv_lt_inv h
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
specialize le_quotient (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)) gamma_star_neq gamma_star d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
have le_prod : ∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ∞ → a < b → a * c < b * c :=
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
by
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
intro a b c hc_nn hc_finite ha d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))a, b, c : ℝ≥ 0 ∞ hc_nn : c ≠ 0 hc_finite : c ≠ ⊤ ha : a < b
a * c < b * c
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
exact Iff.mpr (ENNReal.mul_lt_mul_right hc_nn hc_finite) ha
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
have enn_KSD_neq : ENNReal.ofReal (KSD (μ_t t) π) ≠ 0 := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))le_prod : ∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))le_prod : ∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))le_prod : ∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))le_prod : ∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * c
ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
exact Iff.mp zero_lt_iff (gt_trans (KSD_bounded t) γ_nn)
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
have enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ∞ := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))le_prod : ∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * cenn_KSD_neq : ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))le_prod : ∀ (a b c : ℝ≥ 0 ∞ ), c ≠ 0 → c ≠ ⊤ → a < b → a * c < b * cenn_KSD_neq : ENNReal.ofReal (KSD (μ_t t) π) ≠ 0
ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
simp [Meta.Tactic.simp.rewrite] @ ne_eq:1000 , ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤ ==> ¬ ENNReal.ofReal (KSD (μ_t t) π) = ⊤
[Meta.Tactic.simp.rewrite] @ ENNReal.ofReal_ne_top:1000 , ENNReal.ofReal (KSD (μ_t t) π) = ⊤ ==> False
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000 , ¬ False ==> True
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
specialize le_prod (1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)))) (1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))) (ENNReal.ofReal (KSD (μ_t t) π)) enn_KSD_neq enn_KSD_finite le_quotient d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ t : ℝ≥ 0 log_Sobolev : KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) gamma_star : γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t) le_quotient : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))enn_KSD_neq : ENNReal.ofReal (KSD (μ_t t) π) ≠ 0 enn_KSD_finite : ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤ le_prod : 1 / (2 * ((fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t))) *
ENNReal.ofReal (KSD (μ_t t) π) <
1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤
∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
calc KL (μ_t t) (dμ_t t) dπ = 1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π) := log_Sobolev
_ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π) := le_prod
}
-- We use the previous inequality and the Gronwall's lemma to show that ∀t, KL(μ_t || π) ≤ KL (μ_0 || π) * exp(-2t γ/2KL(μ_0 || π)).
have bound_gronwall : ∀ t> 0 , (KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * t)) := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
neg.h.intro.intro.intro False
by
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
intro t pos_t d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
-- The lemma *ennreal_quo_ineq* allows us to transform the inequality KL(μ_t || π) < 1/(2*γ/2KL(μ_0 || π)) * KSD(μ_t | π) into -KSD(μ_t | π) ≤ -2 * γ/2KL(μ_0 || π) * KL(μ_t || π)
have compute_ineq := ennreal_quo_ineq (KL (μ_t t) (dμ_t t) dπ) (ENNReal.ofReal (KSD (μ_t t) π)) ((γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) (d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0
ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤
simp [Meta.Tactic.simp.rewrite] @ ne_eq:1000 , ENNReal.ofReal (KSD (μ_t t) π) ≠ ⊤ ==> ¬ ENNReal.ofReal (KSD (μ_t t) π) = ⊤
[Meta.Tactic.simp.rewrite] @ ENNReal.ofReal_ne_top:1000 , ENNReal.ofReal (KSD (μ_t t) π) = ⊤ ==> False
[Meta.Tactic.simp.rewrite] not_false_eq_true:1000 , ¬ False ==> True ) gamma_star_neq gamma_star_finite (le_of_lt (bounded_log_Sobolev t)) d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - ENNReal.toReal (ENNReal.ofReal (KSD (μ_t t) π)) ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)
∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
rw [ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - ENNReal.toReal (ENNReal.ofReal (KSD (μ_t t) π)) ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
toReal_ofReal (ksd_nn_t t) d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
at compute_ineq d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
-- As d_KL_t t ≤ -KSD (μ_t t | π) and -KSD (μ_t t | π) ≤ -2 * γ/2KL(μ_0 || π) * KL(μ_t t || π), then d_KL_t t ≤ -2 * γ/2KL(μ_0 || π) * KL(μ_t t || π).
have dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal ((γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) := ge_trans compute_ineq (dkl_ksd t)d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
-- We finally can use the Gronwall's lemma with ψ := t ↦ -2 * γ/2KL(μ_0 || π).
specialize gronwall (fun d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
t Warning: unused variable ` t` [linter.unusedVariables] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
↦ - 2 * ENNReal.toReal ((γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))) t pos_t dkl_ineq d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) gronwall : KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ *
exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
-- We rewrite ∫ s ∈ [0, t], -2 * γ/2KL(μ_0 || π) dt as -2t γ/2KL(μ_0 || π).
rwa [ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) gronwall : KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ *
exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
integral_of_constant d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) gronwall : KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
] d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)t : ℝ≥ 0 pos_t : t > 0 compute_ineq : - KSD (μ_t t) π ≤
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ)dkl_ineq : d_KL_t t ≤ - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) gronwall : KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
at gronwall
}
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
have minus_gamma_star_neg : - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0 := d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
neg.h.intro.intro.intro False
by d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0
{ d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0
have gamma_star_nn : 0 < (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) := Iff.mpr zero_lt_iff gamma_star_neqd : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)gamma_star_nn : 0 < γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0
simp [Meta.Tactic.simp.rewrite] @ neg_mul:1000 , - 2 *
ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)) ==> - (2 * ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)))
[Meta.Tactic.simp.rewrite] @ Left.neg_neg_iff:1000 , - (2 * ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ))) <
0 ==> 0 < 2 * ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ))
[Meta.Tactic.simp.rewrite] @ gt_iff_lt:1000 , 0 < 2 ==> 0 < 2
[Meta.Tactic.simp.rewrite] @ zero_lt_two:1000 , 0 < 2 ==> True
[Meta.Tactic.simp.rewrite] @ zero_lt_mul_left:1000 , 0 <
2 * ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)) ==> 0 < ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ))
[Meta.Tactic.simp.rewrite] @ gt_iff_lt:1000 , 0 <
ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)) ==> 0 < ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ)) d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)gamma_star_nn : 0 < γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )
0 < ENNReal.toReal (γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ))
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)
- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0
exact toReal_pos_iff.mpr ⟨gamma_star_nn, Ne.lt_top gamma_star_finite⟩
}
-- As -2 γ/2KL(μ_0 || π) < 0, lim_(t → ∞) exp(-2 γ/2KL(μ_0 || π) t) = 0.
have exp_limit := exp_tends_to_zero (- 2 * ENNReal.toReal ((γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )))) (KL (μ_t 0 ) (dμ_t 0 ) dπ) minus_gamma_star_negd : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)minus_gamma_star_neg : - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0 exp_limit : limit (fun t => KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t))
0
neg.h.intro.intro.intro False
-- Using the squeeze theorem and the previous result, we prove that lim_(t → ∞) KL(μ_0 || π) t) = 0.
have contradiction_limit := squeeze_th_KL dπ μ_t dμ_t (fun t ↦ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * t)) exp_limit bound_gronwalld : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)minus_gamma_star_neg : - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0 exp_limit : limit (fun t => KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t))
0 contradiction_limit : limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
neg.h.intro.intro.intro False
-- The last result implies that l = 0 which is a contradiction as we supposed l ≠ 0.
have lim_eq_zero := limit_equiv (fun t => KL (μ_t t) (dμ_t t) dπ) l 0 ⟨lim, contradiction_limit⟩d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : 0 ≠ llim_nn : 0 < lγ : ℝ≥ 0 ∞ KSD_bounded : ∀ (t : ℝ≥ 0 ), γ < ENNReal.ofReal (KSD (μ_t t) π)γ_nn : 0 < γγ_finite : γ ≠ ⊤ log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ =
1 / (2 * (ENNReal.ofReal (KSD (μ_t t) π) / (2 * KL (μ_t t) (dμ_t t) dπ))) * ENNReal.ofReal (KSD (μ_t t) π)gamma_star : ∀ (t : ℝ≥ 0 ),
γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ) <
(fun t => ENNReal.ofReal (KSD (μ_t t) π)) t / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) t)gamma_star_neq : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ 0 gamma_star_finite : γ / (2 * KL (μ_t 0 ) (dμ_t 0 ) dπ) ≠ ⊤ bounded_log_Sobolev : ∀ (t : ℝ≥ 0 ),
KL (μ_t t) (dμ_t t) dπ < 1 / (2 * (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 ))) * ENNReal.ofReal (KSD (μ_t t) π)bound_gronwall : ∀ t > 0 ,
KL (μ_t t) (dμ_t t) dπ ≤
KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t)minus_gamma_star_neg : - 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) < 0 exp_limit : limit (fun t => KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (- 2 * ENNReal.toReal (γ / (2 * (fun t => KL (μ_t t) (dμ_t t) dπ) 0 )) * ↑ t))
0 contradiction_limit : limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0 lim_eq_zero : l = 0
neg.h.intro.intro.intro False
d : ℕ inst✝¹³ : MeasurableSpace (Vector ℝ d) inst✝¹² : MeasureSpace (Vector ℝ d) inst✝¹¹ : MeasureSpace ℝ μ, π, ν : Measure (Vector ℝ d) dμ, dπ : Vector ℝ d → ℝ≥ 0 ∞ _h1 : absolutely_continuous μ π _h2 : absolutely_continuous π ν hμ : is_density μ ν dμ hπ : is_density π ν dπ mdμ : Measurable dμ mdπ : Measurable dπ hdμ : ∀ (x : Vector ℝ d), dμ x ≠ 0 ∧ dμ x ≠ ⊤ hdπ : ∀ (x : Vector ℝ d), dπ x ≠ 0 ∧ dπ x ≠ ⊤ inst✝¹⁰ : IsProbabilityMeasure μ inst✝⁹ : IsProbabilityMeasure π h_m_set : ∀ (s : Set (Vector ℝ d)), MeasurableSet sH₀ : Set (Vector ℝ d → ℝ) inst✝⁸ : NormedAddCommGroup (Vector ℝ d → ℝ) inst✝⁷ : InnerProductSpace ℝ (Vector ℝ d → ℝ) s : RKHS H₀ h_kernel_positive : positive_definite_kernel μ H₀ H : Set (ℕ → Vector ℝ d → ℝ) inst✝⁶ : NormedAddCommGroup (ℕ → Vector ℝ d → ℝ) inst✝⁵ : InnerProductSpace ℝ (ℕ → Vector ℝ d → ℝ) dk : Vector ℝ d → ℕ → Vector ℝ d → ℝ d_ln_π, dϕ : ℕ → Vector ℝ d → ℝ is_integrable_H₀ : ∀ (f : Vector ℝ d → ℝ), Integrable fd_ln_π_μ : ℕ → Vector ℝ d → ℝ hd_ln_π_μ : (∀ (x : Vector ℝ d) (i : ℕ), d_ln_π_μ i x = 0 ) → ∃ c, ∀ (x : Vector ℝ d), log (dμ x / dπ x) = c dπ' : ℕ → Vector ℝ d → ℝ hπ' : ∀ (x : Vector ℝ d) (i : ℕ), ENNReal.toReal (dπ x) * d_ln_π i x = dπ' i xinst✝⁴ : Norm (Vector ℝ d) KSD : Measure (Vector ℝ d) → Measure (Vector ℝ d) → ℝ ksd_nn : 0 ≤ KSD μ πhstein : SteinClass (ϕ_ μ H₀ dk d_ln_π) dπ hkl_iff : μ = π ↔ KL μ dμ dπ = 0 hkl_diff : μ ≠ π ↔ 0 < KL μ dμ dπ μ_t : ℝ≥ 0 → Measure (Vector ℝ d) dμ_t : ℝ≥ 0 → Vector ℝ d → ℝ≥ 0 ∞ hμ_t : ∀ (t : ℝ≥ 0 ), is_density (μ_t t) ν (dμ_t t)h_prob : ∀ (t : ℝ≥ 0 ), IsProbabilityMeasure (μ_t t)hdμ_t : ∀ (t : ℝ≥ 0 ) (x : Vector ℝ d), dμ_t t x ≠ 0 ∧ dμ_t t x ≠ ⊤ d_KL_t : ℝ≥ 0 → ℝ ksd_nn_t : ∀ (t : ℝ≥ 0 ), 0 ≤ KSD (μ_t t) πinst✝³ : MeasureSpace ℝ≥ 0 inst✝² : NormedAddCommGroup ℝ≥ 0 ∞ inst✝¹ : NormedSpace ℝ ℝ≥ 0 ∞ inst✝ : LocallyFiniteOrder ℝ≥ 0 gronwall : ∀ (ψ : ℝ≥ 0 → ℝ),
∀ t > 0 ,
d_KL_t t ≤ ψ t * ENNReal.toReal (KL (μ_t t) (dμ_t t) dπ) →
KL (μ_t t) (dμ_t t) dπ ≤ KL (μ_t 0 ) (dμ_t 0 ) dπ * exp (∫ (s : ℝ≥ 0 ) in ↑ (Icc 0 t), ψ s)dkl_ksd : ∀ (t : ℝ≥ 0 ), d_KL_t t ≤ - KSD (μ_t t) πkl_decreasing : ∀ (t t' : ℝ≥ 0 ), t < t' → KL (μ_t t') (dμ_t t') dπ ≤ KL (μ_t t) (dμ_t t) dπkl_finite : ∀ (t : ℝ≥ 0 ), KL (μ_t t) (dμ_t t) dπ ≠ ⊤ l : ℝ≥ 0 ∞ lim : limit (fun t => KL (μ_t t) (dμ_t t) dπ) l _lim_pos : 0 ≤ lKL_bounded : ∀ (x : ℝ≥ 0 ), l < (fun t => KL (μ_t t) (dμ_t t) dπ) xhl : ¬ 0 = l
neg limit (fun t => KL (μ_t t) (dμ_t t) dπ) 0
exact hl.symm lim_eq_zero
}